Higher order Glaeser inequalities and optimal regularity of roots of real functions

TL;DR

This paper generalizes Glaeser inequalities to higher orders, providing new estimates on derivatives of roots of functions, which improve understanding of their regularity and optimality of these bounds.

Contribution

It introduces a higher order Glaeser inequality and applies it to establish optimal regularity results for roots of functions with Holder continuous derivatives.

Findings

Derived a higher order Glaeser inequality.

Established pointwise derivative estimates for roots of functions.

Proved the optimality of the regularity results.

Abstract

We prove a higher order generalization of Glaeser inequality, according to which one can estimate the first derivative of a function in terms of the function itself, and the Holder constant of its k-th derivative. We apply these inequalities in order to obtain pointwise estimates on the derivative of the (k+alpha)-th root of a function of class C^{k} whose derivative of order k is alpha-Holder continuous. Thanks to such estimates, we prove that the root is not just absolutely continuous, but its derivative has a higher summability exponent. Some examples show that our results are optimal.